The Landau Hamiltonian with δ-potentials supported on curves

Abstract

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα = (i∇ + A) 2 + αδΣ in L 2 (R 2 ) with a δ-potential supported on a finite C 1,1 -smooth curve Σ are studied. Here A = 1 2 B(−x2, x1) ⊤ is the vector potential, B > 0 is the strength of the homogeneous magnetic field, and α ∈ L ∞ (Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ. After a general discussion of the qualitative spectral properties of Aα and its resolvent, one of the main objectives in the present paper is a local spectral analysis of Aα near the Landau levels B(2q + 1), q ∈ N0. Under various conditions on α it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of α. Furthermore, the use of Landau Hamiltonians with δ-perturbations as model operators for more realistic quantum systems is justified by showing that Aα can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.Organization of the paper. Section 2 contains some preliminary material concerning the unperturbed Landau Hamiltonian, properties of Schatten-von Neumann ideals and some aspects of perturbation theory. In Subsection 2.4 we discuss a class of Toeplitz-like operators related to Landau Hamiltonians. In Section 3 we make use of the abstract concept of quasi boundary triples and their Weyl functions (see Appendix A for a brief introduction) in order to study Landau Hamiltonians with δ-potentials supported on curves. Using a suitable quasi boundary triple we show self-adjointness of A α , provide qualitative spectral properties, and derive the Krein-type resolvent formula (1.4). The approximation of A α by magnetic Schrödinger operators with scaled regular potentials is also discussed; the proof is technical and therefore outsourced to Appendix B. Section 5 is devoted to the spectral analysis of the compressed resolvent difference P q W λ P q . Under various assumptions we obtain spectral estimates and spectral asymptotics for this operator. Based on these results we provide our main results on the eigenvalue clusters of A α at Landau levels in Section 6.

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

“…(3.9) It is well known that Φ λ and C λ are bounded and everywhere defined and that 11) and this operator is also bounded when viewed as an operator from L 2 (Ω; C 4 ) to H 1/2 (∂Ω; C 4 ); cf. [12, equation (2.12) and the discussion below].…”

“…For that, we use the following result on operators with range in the Sobolev space . Its proof follows word-by-word the one of [ 11 , Proposition 2.4]; hence, we omit it here.…”

“…Our mathematical treatment of the operators in ( 1.1 ) is based on the application of a suitable so-called quasi boundary triple. Quasi boundary triples and their Weyl functions are an abstract concept from extension and spectral theory for symmetric and self-adjoint operators which were originally introduced to investigate boundary value problems for elliptic partial differential operators in [ 17 ], but proved to be useful in many other situations, see, e.g., [ 11 , 19 , 20 ]. Quasi boundary triples were also applied more recently in [ 10 , 15 ] to Dirac operators with singular potentials.…”

The fate of Landau levels under $δ$-interactions

“…the singular interaction does not have an effect on u, and hence H υ u = H 0 u = Λ q u. This allows one to show with the help of [7,Lemma 3.7] that the kernel of T q (υδ Γ ) is finite-dimensional under the assumption that υ is strictly positive. Moreover, this connection provides a direct link to nodal sets for eigenfunctions of H 0 and the non-emptiness of ker (H υ − Λ q ).…”

“…In the spectral gaps (Λ q−1 , Λ q ), where q ∈ Z + and Λ −1 := −∞, of H 0 there may appear discrete eigenvalues of H υ which can only accumulate at the Landau levels Λ q , q ∈ Z + . Some results on the asymptotic distribution near any fixed Λ q of these discrete eigenvalues were obtained in [7]. In particular, it was shown that if either υ ≥ 0 or υ ≤ 0 on Γ, v ≡ 0, and certain additional regularity assumptions hold, then in a neighbourhood of any Λ q there are infinitely many discrete eigenvalues of H υ and their accumulation rate to the Landau levels is described in terms of the logarithmic capacity of the interaction support; cf.…”